Gravity is the force of inherent natural attraction between two massive bodies. The magnitude of the gravitational force is directly related to the mass of the bodies and is inversely related to the square of the distance between centers of mass of the two attracted bodies.
Gravity is measured as acceleration, g, usually as a vertical vector component. The freefall acceleration, g, of an object near the surface of the earth is given to a first approximation by the gravitational attraction of a point with the mass of the entire earth, Me, located at the center of the earth, a distance, Re, from the surface of the earth. This nominal gravity value, g=G×Me/Re2, is about 9.8 m/s2. The nominal gravity value varies over the earth's surface in a relatively small range of about 0.5%. At the equator, the nominal gravity value is about 9.780 m/s2, and at the north and south poles, the nominal gravity value is about 9.830 m/s2. The common unit of measurement for gravity is the “Galileo” (Gal), which is a unit of acceleration defined as 1 cm/s2. One Gal generally approximates 1/1000 (10−3) of the force of gravity at the earth's surface. An instrument used to measure gravity is called a “gravimeter.”
Gravity measurements are used in a number of practical applications, such as mapping subsurface geology, exploration and development of mineral and hydrocarbon resources, volcanology, geotechnical investigations and the environment. Subsurface gravity measurements are typically made by lowering a gravimeter in a borehole or well bore, and measuring gravity at intervals along the depth of an area of interest. With sufficiently accurate gravity measurements at multiple intervals, the mean bulk density of the formation of interest can be determined. Mean bulk density information is particularly useful to monitor the change in condition of hydrocarbon reservoirs which contain crude oil or natural gas. The mean bulk density information indicates the extent to which the crude oil or natural gas has flowed from or changed in position within the formation. This information is useful to optimize the efficiency of extraction of these hydrocarbon resources.
To be the most effective in determining mean bulk density, the gravity measurements should have accuracy to at least nine digits (10−9). Measurements of lesser accuracy may not be adequate to obtain a meaningful evaluation of mean bulk density. In addition, the gravimeter itself must have a physical size which allows it to be inserted into and moved along a borehole. A diameter of a borehole in which gravity measurements are taken may be as small as 2⅜ inches, although the borehole of a producing well in which gravity measurements are also taken is typically larger.
At the present time, a differential interferometric gravimeter has the capability of gravity measurement accuracy to at least nine digits. An example of this type of gravimeter is described in U.S. Pat. No. 5,892,151, invented by the inventor hereof. In general, a differential interferometric gravimeter uses at least one test mass which is released to fall freely under the influence of gravity within a vacuum chamber, while a laser beam impinges upon and reflects in at least two separate beams from the freely falling test mass. The two beams are combined, and phase differences in the two combined light beams create interference fringes. The interference fringes correlate to the gravity value. This type of differential interferometric gravimeter involves complex and sensitive equipment, and is prone to adverse influences from environmental perturbations. In addition, the physical size of the equipment is considerably larger than the size of a typical borehole. For these and other reasons, a differential interferometric gravimeter is not suitable for measuring gravity in small diameter boreholes.
Another type of gravimeter is a relative gravimeter. In general, a relative gravimeter suspends a test mass from a spring or other type of suspension device, and then measures the extent to which the change in gravity alters the extent of elongation of the spring or suspension device. While relative gravimeters are small in size and capable of fitting within a typical borehole, the accuracy of measurement is not remotely close to nine digit accuracy.
Another type of gravimeter utilizes a pendulum to measure gravity. A mass or “bob” is suspended by an arm that is connected to a point of suspension or center of motion. Energy imparted to the bob causes it to swing back and forth in an arc of oscillation. Gravity sustains the oscillation of the bob until the inherent friction of mechanical movement dissipates the energy initially imparted to move the bob.
The time required for the pendulum bob to execute one oscillation or swing from one point in the arc of oscillation back to that same point is the period (T) of oscillation. The period (T) of oscillation, the value of gravity (g) and the length of the pendulum arm (L) are related to one another by the following well-known equation (1):T=2π[L/g]1/2  (1)
From equation (1), it is apparent that the value of gravity is related to the length of the pendulum arm (L) and inversely related to the period (T) of oscillation. By measuring the length of the pendulum arm (L) and by measuring the period (T) of oscillation, the value of gravity is determined by the following equation (2), which is a rearranged version of equation (1):g=4π2L/T2  (2)
Alternatively, since the frequency (f) of oscillatory movement is the inverse of the period of oscillatory movement, the frequency (f) of oscillation of the bob is equal to 1/T. Applying this to equation (2) shows that the value of gravity is also related to the frequency (f) of the pendulum oscillation by the following equation (3):g=4π2Lf2  (3)
Because there are no theoretical limits on the length (L) of the pendulum arm or on the period (T) or the frequency (f) of the oscillatory movement, the pendulum itself can be made sufficiently small so that it can be inserted within a typical borehole and be used to measure gravity values at intervals within the borehole. Although the pendulum solves the size problems for gravity measurement in small diameter boreholes, certain other practical problems arise. These problems center around the practical recognition that actual performance of a pendulum departs from the ideal or theoretical behavior defined by equations (1)-(3) and also around the practical difficulty of operating the pendulum in an outside environment as compared to a closely controlled laboratory environment.
Equations (1)-(3) do not predict the ideal behavior of a pendulum when the arc of oscillation departs from an infinitesimally small angle. A practical and workable pendulum must have an arc of oscillation which is greater than an infinitesimally small angle, and in which case, the period (T) or the frequency (f) becomes dependent upon the arc of oscillation. The arc of oscillation is also related to the maximum amplitude points of the bob during oscillation. A greater arc of oscillation results in greater maximum amplitude points. The maximum amplitude points are measured transversely from to a vertical reference through the point of suspension.
A mathematical correction factor can be applied to correct the period (T) or frequency (f) based on the arc of oscillation or the maximum amplitude points during oscillation, when those values are greater than an infinitesimal value. This mathematical correction factor is a complicated expansion of a elliptical integral in a power series, and is described in “The Earth and Its Gravity Field,” by Heiskanen and Meines, McGraw-Hill, 1958, pp. 87-93. The necessity to calculate and apply a mathematical correction factor complicates the gravity measurement.
Another practical difficulty is that the pendulum will not continue to oscillate indefinitely, due to the loss of oscillation energy caused by frictional movement of the swinging pendulum. The loss of energy has the practical effect of continually decreasing the arc of oscillation and the maximum amplitude points. It is more difficult to measure the arc of oscillation and the maximum amplitude points of oscillation under the circumstances of decreasing or decaying oscillation of the pendulum. Without an accurate measurement of these values, an accurate determination of the value of gravity is not possible.
It is possible to calculate a gravity value while the pendulum is undergoing a decay in the angle of oscillation due to energy loss. However, since the correction factor depends on the angle of oscillation, the correction factor must be recalculated and reapplied continuously as the arc of oscillation and the maximum amplitude points decrease during decaying oscillation. The calculations must be continually coordinated with measurements of the period (T) or frequency (f), as the oscillation of the pendulum decays. Considerable computation is required to derive a gravity value under these circumstances, and even then, the accuracy may be compromised due to the difficulty in measuring the continually changing values required to derive an accurate gravity value determination.
It theoretically possible to add energy to the pendulum to counteract the frictional energy loss and thereby establish and maintain a constant arc of oscillation with constant maximum amplitude points of oscillation. However, adding energy to the pendulum substantially increases the risk of disturbing the normal oscillatory motion and creating unwanted modes of motion by the pendulum bob. Unwanted modes of motion and the energy which creates those unwanted modes of motion adversely influence oscillation and cause the pendulum to depart from a desired plane of oscillation. The oscillatory movement outside of the desired plane of oscillation does not accurately represent the effect of gravity in the measured plane of oscillation, thereby introducing errors in the gravity values determined.
To avoid the disruptive effects of adding energy to an oscillating pendulum, a finite amount of energy may be added to the pendulum followed by an interval where any disruptive effects of the added energy are expected to dissipate. Theoretically, a sufficient amount of energy can be added so that the disruptive effects of the energy addition will have dissipated when the oscillation motion decays through a preselected maximum amplitude point and arc of oscillation. While this approach of adding energy does have the theoretical effect of more closely achieving a predetermined arc of oscillation and points of maximum amplitude where instantaneous measurements can be taken, the approach involves risks of inaccuracy in measurement of the values required. The time required to accomplish sequential gravity value measurements is also extended due to the necessity to allow the disruptive effects of the energy addition to dissipate.
Seismic noise is particularly troublesome problem in attempting to make accurate gravity measurements using a pendulum gravimeter. Seismic noise is a naturally occurring phenomenon resulting from natural movement of the earth itself and its subsurface formations. Seismic noise occurs continuously and has random, unpredictable intensity. The magnitude of normal seismic noise is so small that it is not humanly perceptible. However, the magnitude of normal seismic noise is sufficient to adversely affect the arc of oscillation and maximum amplitude points of a pendulum, making the measurement of gravity (g) with the accuracy of nine digits (10−9) impossible with a single pendulum gravimeter.